Finite-difference and finite-volume methods for nonlinear standing ultrasonic waves in fluid media

In the framework of the application of high-power ultrasonics in industrial processing in fluid media, the mathematical prediction of the acoustical parameters inside resonators should improve the development of practical systems. This can be achieved by the use of numerical tools able to treat the nonlinear acoustics involved in these phenomena. In particular, effects like nonlinear distortion and nonlinear attenuation are fundamental in applications. In this paper, three one-dimensional numerical models in the time domain for calculating the nonlinear acoustic field inside a one-dimensional resonant cavity are presented and compared. They are based on the finite-difference and the finite-volume methods. These different algorithms solve the differential equations, from the linear up to the strongly nonlinear case (including weak shock). Some physical results obtained from the modelling of ultrasonic waves and a comparison of the efficiency of the different algorithms are presented.     

谐振管内非线性驻波的有限体积数值算法

为了扩展谐振管内非线性驻波在工程中的应用, 以及克服现有数值计算方法仅局限于求解直圆柱形和指数形谐振管内非线性驻波的问题. 根据变截面的非稳态可压缩热黏性流体Navier-Stokes方程和空间守恒方程, 并基于求解压力速度耦合方程的半隐式算法和交错网格技术, 构建一种能够计算任意形状轴对称谐振管受活塞驱动时内部非线性驻波的有限体积算法. 分别对圆柱形、指数形和圆锥形谐振管内的非线性驻波进行仿真计算. 通过与现有试验结果以及数值仿真结果的对比, 验证了该方法的正确性.并获得除驻波声压之外的另外一些新的物理结果, 包括速度、密度、温度的瞬时变化.在直圆柱形谐振管内产生冲击声压波, 速度波形中出现钉状结构.而在指数形和圆锥形谐振管内产生高声压幅值的驻波, 没有出现冲击波, 速度波形中均未发现钉状结构. 计算结果表明谐振管内非线性驻波的物理属性与谐振管形状之间有密切关系.     

Numerical and experimental analysis of strongly nonlinear standing acoustic waves in axisymmetric cavities

In this paper the behaviour of strongly nonlinear waves in axisymmetric resonators is experimentally and numerically studied. Experiments are carried out in a cylindrical cavity, which transversal dimension is bigger than the longitudinal one, excited by a narrow band transducer. The quality factor and displacement amplitudes are experimentally quantified. A finite difference numerical model is developed to solve, in the time domain, a proposed set of full nonlinear differential equations written in Lagrangian coordinates. Pressure field is obtained for complicated modes. Good agreement between numerical and experimental results is found. New nonlinear properties of quasi-standing waves in axisymmetric resonators are described. Results are compared to linear approximation and show the importance of three-dimensional analysis.     

A numerical formulation for nonlinear ultrasonic waves propagation in fluids

A finite-difference algorithm is developed for analysing the nonlinear propagation of pulsed and harmonic ultrasonic waves in fluid media. The time domain model allows simulations from linear to strongly nonlinear plane waves including weak shock. Effects of absorption are included. All the harmonic components are obtained from only one solving process. The evolution of any original signal can be analysed. The nonlinear solution is obtained by the implicit scheme via a fast linear solver. The numerical model is validated by comparison to analytical data. Numerical experiments are presented and commented. The effect of the initial pulse shape on the evolution of the pressure waveform is especially analysed.     

Nonlinear ultrasonic waves in bubbly liquids with nonhomogeneous bubble distribution: Numerical experiments

This paper deals with the nonlinear propagation of ultrasonic waves in mixtures of air bubbles in water, but for which the bubble distribution is nonhomogeneous. The problem is modelled by means of a set of differential equations which describes the coupling of the acoustic field and bubbles vibration, and solved in the time domain via the use and adaptation of the SNOW-BL code. The attenuation and nonlinear effects are assumed to be due to the bubbles exclusively. The nonhomogeneity of the bubble distribution is introduced by the presence of bubble layers (or clouds) which can act as acoustic screens, and alters the behaviour of the ultrasonic waves. The effect of the spatial distribution of bubbles on the nonlinearity of the acoustic field is analyzed. Depending on the bubble density, dimension, shape, and position of the layers, its effects on the acoustic field change. Effects such as shielding and resonance of the bubbly layers are especially studied. The numerical experiments are carried out in two configurations: linear and nonlinear, i.e. for low and high excitation pressure amplitude, respectively, and the features of the phenomenon are compared. The parameters of the medium are chosen such as to reproduce air bubbly water involved in the stable cavitation process.     

Numerical simulations of three-dimensional nonlinear acoustic waves in bubbly liquids

This paper presents three-dimensional simulations of nonlinear propagation of ultrasonic waves through bubbly liquids, which represent the continuity of our previous works included in the numerical tool SNOW-BL. The behavior of three-dimensional nonlinear acoustic waves in bubbly liquids is analyzed by means of numerical predictions. Nonlinearity, attenuation, and dispersion due to the presence of bubbles in the liquid are taken into account. The numerical solution to the differential problem is obtained by means of a finite-difference scheme. The simulations we present here consider a homogeneous distribution of bubbles in the liquid. Results compare high and low-amplitude waves to detect the nonlinear effects of the bubbles. Results are shown for radiation and enclosure problems.     

Spatio-temporal development of the filaments due to the thermal self-focusing instability near the critical surface in ionospheric plasmas

We study fully nonlinear spatio-temporal development of the thermal self-focusing instability of high-power radio waves near the critical surface of the ionosphere. These simulations improve on our earlier work by including an evolution equation for the density instead of using the assumption of constant pressure to determine the perturbed density connected with the known temperature perturbation. Using our two-dimensional nonlinear code we analyze the time scale and associated velocity for the development of the field-aligned irregularities as they spread from the critical surface both in the underdense as well as the overdense regions. The scaling of this velocity as a function of the radiated power of the heater electromagnetic wave (ERP) is determined. We also study the characteristic size of the self-focused filament as a function of ERP. Finally, the spectrum of the density and temperature fluctuations as well as modifications in the equilibrium values of these parameters for different values of ERP are presented. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 7, pp. 670–681, July 1999.     

Acoustic cavitation mechanism: a nonlinear model

During acoustic cavitation process, bubbles appear when acoustic pressure reaches a threshold value in the liquid. The ultrasonic field is then submitted to the action of the bubbles. In this paper we develop a model to analyze the cavitation phenomenon in one-dimensional standing waves, based on the nonlinear code SNOW-BL. Bubbles are produced where the minimum rarefaction pressure peak exceeds the cavitation threshold. We show that cavitation bubbles appear at high amplitude and drastically affect (dissipation, dispersion, and nonlinearity) the ultrasonic field. This paper constitutes the first work that associates the nonlinear ultrasonic field to a bubble generation process.     

Review on second-harmonic generation of ultrasonic guided waves in solid media(Ⅰ): Theoretical analyses

Considering the high sensitivity of the nonlinear ultrasonic measurement technique and great advantages of the guided wave testing method, the use of nonlinear ultrasonic guided waves provides a promising means for evaluating and characterizing the hidden and/or inaccessible damage/degradation in solid media. Increasing attention on the development of the testing method based on nonlinear ultrasonic guided waves is largely attributed to the theoretical advances of nonlinear guided waves propagation in solid media. One of the typical acoustic nonlinear responses is the generation of second harmonics that can be used to effectively evaluate damage/degradation in materials/structures. In this paper, the theoretical progress of second-harmonic generation(SHG) of ultrasonic guided wave propagation in solid media is reviewed. The advances and developments of theoretical investigations on the effect of SHG of ultrasonic guided wave propagation in different structures are addressed. Some obscure understandings and the ideas in dispute are also discussed.     

Numerical simulation of three-dimensional nonlinear water waves

We present an accurate and efficient numerical model for the simulation of fully nonlinear (non-breaking), three-dimensional surface water waves on infinite or finite depth. As an extension of the work of Craig and Sulem [19], the numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing the Dirichlet–Neumann operator which is described in terms of its Taylor series expansion in homogeneous powers of the surface elevation. Each term in this Taylor series can be computed efficiently using the fast Fourier transform. An important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the Hamiltonian equations of motion, as well as detailed numerical tests on the convergence of the Dirichlet–Neumann operator. The performance of the model is illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water.     

Review on second-harmonic generation of ultrasonic guided waves in solid media (I):Theoretical analyses

Considering the high sensitivity of the nonlinear ultrasonic measurement technique and great advantages of the guided wave testing method, the use of nonlinear ultrasonic guided waves provides a promising means for evaluating and characterizing the hidden and/or inaccessible damage/degradation in solid media. Increasing attention on the development of the testing method based on nonlinear ultrasonic guided waves is largely attributed to the theoretical advances of nonlinear guided waves propagation in solid media. One of the typical acoustic nonlinear responses is the generation of second harmonics that can be used to effectively evaluate damage/degradation in materials/structures. In this paper, the theoretical progress of second-harmonic generation (SHG) of ultrasonic guided wave propagation in solid media is reviewed. The advances and developments of theoretical investigations on the effect of SHG of ultrasonic guided wave propagation in different structures are addressed. Some obscure understandings and the ideas in dispute are also discussed.     

条状障碍物对超声非线性声场的影响研究

基于Khokhlov-Zabolotkaya-Kuznetsov(KZK)方程，在频域建立了求解三维非轴对称声场的方法，理论及实验研究了声波通过肋骨条状障碍物后的非线性声场分布，数值计算与实验结果相符.本文还讨论了条状障碍物对空间平均-时间平均声强(I_SATA)的影响. 关键词： 非轴对称 频域法 肋骨 衍射场     

Effect of resonator dimensions on nonlinear standing waves

An investigation of the effect of resonator dimensions on nonlinear standing waves in shaped resonators is conducted. Simple forms of the shear viscosity term in the momentum equations are developed for an axisymmetric (2D) resonator and a low aspect ratio rectangular (3D) resonator. The cross sections of the resonators are exponentially expanded and the one-dimensional wave equations are solved by using the Galerkin's method. The quality factors, pressure waveforms, compression ratios, and resonance frequencies are calculated for different dimensionless cross sections and lengths of the resonators. The results show that, apart from the resonator length, the ratio of the cross-section dimension to the length of the resonator is an important parameter. If the ratio is greater than 0.04, the characteristics of the shaped resonator are not affected significantly. However, when the ratio is less than 0.01, the resonance becomes weak, the compression ratio drops substantially, and the frequency response changes as well.     

Physics and mechanism of ultrasonic impact

More and more experts and researchers in industry express their interest in the application of deformation effects of various peening techniques on the metal surface. This is primarily due to a relatively simple directional change in condition at the surface and in sub-surface layers of the material as a result of plastic deformation due to impulses of force caused, among other things, by converting ultrasonic oscillations of various impacting elements (indenters) at the treated surface. These effects are of a stochastic nature and their duration (or the time of impact) is generally measured in units of microseconds. To obtain relatively uniform coverage, an operator may use several treatment passes. However, a stochastic nature of single impacts makes it difficult to obtain a uniform distribution of deformations and hence surface characteristics as specified, in particular, by the engineering standards. We have developed the methods and means of implementing the ultrasonic impact and controlling its parameters. A fundamental distinction of the ultrasonic impact is that its duration is measured in the range from hundreds of microseconds to units of milliseconds, while the parameters responsible for the effects upon the surface may be adjusted according to the task. It is important to note that in the frequency range of processing ultrasound of up to 80 kHz this feature of the ultrasonic impact allows utilizing the plastic deformation region as a matched membrane to transmit ultrasonic oscillations and excite ultrasonic stress waves in the material being treated. These phenomena, in turn, initiate highly effective relaxation processes, plastic deformation and, as a result thereof, effects upon the structure and properties of the material, which are adequate to the task. This paper describes the theory and the results of the experimental investigations into the physics of the ultrasonic impact. Also, the mechanism of the ultrasonic impact implementation based on high-power ultrasonic transducers is addressed. The paper is aimed at engineers and researchers in the area of industrial application of high-power ultrasonics.     

Physical investigation of acoustic waves induced by the oscillation and collapse of the single bubble

The objective of this paper is to apply both experimental and numerical methods to investigate acoustic waves induced by the oscillation and collapse of a single bubble. In the experiments, the schlieren technique is used to capture the temporal evolution of the bubble shapes, and the corresponding acoustic waves. The results are presented for the single bubble generated by a low-voltage bubble generator in the free field of water. During the numerical simulations, a three-dimensional (3D) weakly compressible model is introduced to investigate the single bubble dynamics, including the generation and propagation of acoustic waves. The results show that (1) Compression wave, rarefaction wave and shock wave are generated during expansion stage, collapse stage and rebound stage of the bubble respectively. (2) Compression waves are induced by the rapid expansion of the bubble and eventually steepen into one shock wave propagating outward in the liquid, then another strong shock wave is emitted at the final collapse stage. The velocity and pressure of the liquid field increases after the shock wave. (3) Rarefaction waves are generated during the collapse stage due to the contraction of the bubble. The rarefaction wave reduces the liquid pressure and its spatial distribution is dispersive. The pressure of these acoustic waves and their effect on the liquid velocity attenuate with the increase of propagation distance.     

Viability of yeast cells in well controlled propagating and standing ultrasonic plane waves

Recent studies have shown that there is no loss of cell viability when the cells are subjected to ultrasonic standing wave fields in acoustic cell retention systems. These systems are characterised by waves that spatially vary in pressure amplitude in the direction of sound propagation. In this work an anechoic 'one-dimensional' sonication chamber has been developed that produces propagating waves, which differ from standing waves in that the pressure amplitude remains constant as the wave travels in a medium with negligible attenuation. The viability of yeast cell suspensions as a function of treatment time was investigated during exposure to both standing and propagating wave fields with frequencies slightly above 2 MHz. The influence of 12% (vol/vol) of ethanol in water on the spatial arrangement of the cells in suspension was also studied. Changes in yeast cell morphology caused by the different types of suspension media and the ultrasonic treatment were examined by transmission electron microscopy (TEM). The agglomeration of yeast cells within the pressure nodal planes appears to minimise damaging effects due to ultrasonic fields.     

Modelling for the robust design of layered resonators for ultrasonic particle manipulation

Several approaches have been described for the manipulation of particles within an ultrasonic field. Of those based on standing waves, devices in which the critical dimension of the resonant chamber is less than a wavelength are particularly well suited to microfluidic, or "lab on a chip" applications. These might include pre-processing or fractionation of samples prior to analysis, formation of monolayers for cell interaction studies, or the enhancement of biosensor detection capability. The small size of microfluidic resonators typically places tight tolerances on the positioning of the acoustic node, and such systems are required to have high transduction efficiencies, for reasons of power availability and temperature stability. Further, the expense of many microfabrication methods precludes an iterative experimental approach to their development. Hence, the ability to design sub-wavelength resonators that are efficient, robust and have the appropriate acoustic energy distribution is extremely important. This paper discusses one-dimensional modelling used in the design of ultrasonic resonators for particle manipulation and gives example of their uses to predict and explain resonator behaviour. Particular difficulties in designing quarter wave systems are highlighted, and modelling is used to explain observed trends and predict performance of such resonators, including their performance with different coupling layer materials.     

Three-dimensional hybrid model for predicting air-coupled generation of guided waves in composite material plates

For contact-less, non-destructive testing (NDT) purposes using air-coupled ultrasonic transducers, it is often required to numerically simulate the propagation of ultrasonic waves in solid media, and their coupling through air with specific transducers. At that point, one could simulate the propagation in the air and then in the solid component, using a Finite Element (FE) model. However, when three-dimensional (3D) modeling becomes necessary, such a solution reveals to be extremely demanding in terms of number of degrees of freedom and computational time. In this paper, to avoid such difficulties, the propagation in air from an ultrasonic transmitter to a tested solid plate is modeled in 3D using a closed-form solution. The knowledge of the transducer characteristics (diameter, frequency bandwidth, efficiency in Pa/V) allows the spatial distribution and actual pressure (in Pa) of the acoustic field produced in the air to be predicted, for a given input voltage. This pressure field is applied in turn as a boundary condition in a 3D FE model, to predict the plate response (displacement and stress guided beams) for a given distance between the transmitter and the plate, and for a given angle of orientation of the transmitter with respect to the plate. The FE model is so restricted to modeling of the solid structure only, thus reducing very significantly the number of degrees of freedom and computational time. The material constituting the plate is considered to be an anisotropic and viscoelastic medium. To validate the whole modeling process, an air-coupled ultrasonic transducer is used and oriented at a specific angle chosen for generating one specific Lamb mode guided along a composite plate sample, and a laser probe measures the normal velocity at different locations on the surface of the plate. In the field of NDT, it is generally suitable to excite a pure Lamb mode in order to ease the interpretation of received signals that would represent waves scattered by defects. After a validation step, the numerical model is then used to investigate the effect of the material anisotropy on the purity of the incident guided mode.     

Numerical characterizations of unstable optical resonators and evaluation of the geometry effects

Unstable resonators have been widely used in high-power gas lasers as well as solid-state lasers. The phase and the spatial distribution of intensity of these lasers are very important in some applications such as material processing. In this paper, unstable resonators with three different geometries have been characterized numerically and the results have been compared and evaluated. Based on the Fresnel-Kirchhoff integral, the two-dimensional phase and intensity have been calculated for three different positive branch unstable resonators. The results show that the resonators with rectangular geometry have the best performance for near-field as well as far-field intensity, which is more suitable for material processing. The calculations also show that the maximum output power can be extracted from the rectangular resonator with spherical surface, while the circular resonator with spherical surface has minimum output power. The results also show that the laser has a higher divergence for the cylindrical resonator in compare with those for the circular and rectangular resonators.